Proves strong integrality for colored HOMFLYPT invariants.
problem Proving strong integrality for colored HOMFLYPT invariants.
method Using HOMFLY skein theory.
result Strong integrality theorem for reduced colored HOMFLYPT invariants.
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48 results for “colored HOMFLYPT invariants”
An invariant for knots with colored bonds respects HOMFLYPT relation.
problem Modeling and distinguishing knots with colored bonds.
method Introducing a HOMFLYPT skein module for colored bonded knots.
result The non-rigid version of the module provides information about knottedness of bonds.
The aim of this paper is two-fold. First, we give a fully geometric description of the HOMFLYPT homology of Khovanov-Rozansky. Our method is to construct this invariant in terms of the cohomology of various sheaves on certain algebraic groups, in the same spirit as the authors' previous work on Soergel bimodules. All t…
In this paper, we investigate the properties of the full colored HOMFLYPT invariants in the full skein of the annulus . We show that the full colored HOMFLYPT invariant has a nice structure when . The composite invariant is a combination of the full colored HOMFLYPT invariants. In order to …
Computes colored HOMFLYPT invariants using holomorphic curves.
problem Counting holomorphic curves in Calabi-Yau 3-folds.
method Computes contributions of multiple covers of holomorphic annuli.
result Agrees with topological string theory predictions and proves Ooguri-Vafa formula.
New invariants of links are constructed using the skein invariant polynomial of colored links defined by the author in [1]. These invariants are stronger than the homflypt polynomial.
Categorifies a skein relation for links colored by one-column Young diagrams.
problem Categorification of a skein relation for links.
method Homotopy equivalence between two complexes of singular Soergel bimodules.
result Categorification of the colored skein relation.
This paper discuss an intrinsic relation among congruent relations \cite{CLPZ}, cyclotomic expansion and Volume Conjecture for invariants. Motivated by the congruent relations for invariants obtained in our previous work \cite{CLPZ}, we study certain limits of the invariants at various roots of …
Proves colored HOMFLYPT polynomial is -holonomic.
problem Proving the -holonomic property of colored HOMFLYPT polynomials.
method Skew Howe duality and Poincare-Birkhoff-Witt computation of quantum group.
result Existence of super-polynomial for knots.
New invariant for links in handlebodies defined using braid groups and Soergel bimodules.
problem Defining invariants for links in handlebodies.
method Using braid groups and complexes of Soergel bimodules.
result Generalized HOMFLYPT homology for links in handlebodies.
By generalizing the Kuperberg sl(3) bracket, we construct a graph-valued analogue of the Homflypt sl(3) invariant for virtual knots. The restriction of this invariant for classical knots coincides with the usual Homflypt sl(3) invariant, and for virtual knots and graphs it provides new information that allows one to pr…
Researchers create functors to match colored homologies of knots and links.
problem Equivalence of colored HOMFLYPT homologies for links and knots.
method Constructing functors on singular Soergel bimodules to identify homologies.
result Established parity results for intrinsic column-colored homology of positive torus knots.
New categorified homology expressions for torus knots and links.
problem Computing homology for colored torus knots and links.
method Recursive construction of categorified Young symmetrizers and comparison with row-colored homology.
result Verification of mirror symmetry conjectures for positive torus knots.
Study connects knot contact homology to Chern-Simons theory's large N limit.
problem Relate knot contact homology to Chern-Simons theory's large N limit.
method Prove conjecture linking augmentation varieties to Chern-Simons theory's large N limit; characterize HOMFLYPT difference module.
result Classical limit of HOMFLYPT difference module equals degree 0 abelianized knot contact homology.
Link invariants explained using HOMFLYPT polynomials and Yokonuma-Hecke algebras.
problem Link invariants constructed using Markov traces on Yokonuma-Hecke algebras.
method Description of link invariants in terms of HOMFLYPT polynomials and linking matrix.
result Link invariants explained completely using HOMFLYPT polynomials and Yokonuma-Hecke algebras.
The HOMFLYPT polynomial yields a positive function for a specific group.
problem Constructing representations of the oriented Thompson group.
method Using the HOMFLYPT polynomial as a link invariant.
result The HOMFLYPT polynomial defines a positive definite function of the oriented Thompson group.
Adjusts Yang-Baxter operators for HOMFLYPT polynomials.
problem Computing 2-cocycle invariants for links.
method Adjusts Yang-Baxter operators and computes second homology.
result Potential for 2-cocycle invariant for links.
Paper shows links with same Homflypt but different Conway type invariant.
problem Comparing Homflypt and Conway type invariants for links.
method Applied two skein relations to crossings to define generalized Conway type invariant.
result Found links with same Homflypt but different Conway type invariant.
Algorithm calculates polynomial coefficients of link invariants.
problem Computing first coefficients of link invariants.
method Dynamic programming algorithm for Homflypt and Kauffman polynomials.
result Polynomial time complexity for first coefficients.
Biquandle Virtual Bracketsmath.GT
New quantum invariants for knots and links using biquandle virtual brackets.
problem Quantum enhancements of biquandle counting invariant.
method Defined using skein invariants of biquandle colored diagrams with virtual crossings.
result Includes classical and new invariants, forming an infinite family.
We explore Jaeger's state model for the HOMFLYPT polynomial. We reformulate this model in the language of Gauss diagrams and use it to obtain Gauss diagram formulas for a two-parameter family of Vassiliev invariants coming from the HOMFLYPT polynomial. These formulas are new already for invariants of degree 3.
In this paper we announce the existence of a family of new -variable polynomial invariants for oriented classical links defined via a Markov trace on the Yokonuma-Hecke algebra of type . Yokonuma-Hecke algebras are generalizations of Iwahori-Hecke algebras, and this family contains the Homflypt polynomial, the fa…
We construct graph-valued analogues of the Kuperberg sl(3) and G2 invariants for virtual knots. The restriction of the sl(3) or G2 invariants for classical knots coincides with the usual Homflypt sl(3) invariant and G2 invariants. For virtual knots and graphs these invariants provide new graphical information that allo…
We give formulas expressing Milnor invariants of an n-component link L in the 3-sphere in terms of the HOMFLYPT polynomial as follows. If the Milnor invariant \barμ_J(L) vanishes for any sequence J with length at most k, then any Milnor \barμ-invariant \barμ_I(L) with length between 3 and 2k+1 can be represented as a c…
Paper constructs a HOMFLYPT-type invariant for pseudo links.
problem Inability to construct polynomial invariants for pseudo links using Hecke algebra techniques.
method Using a resolution homomorphism and pseudo Hecke algebra of type \(A\), the paper constructs a HOMFLYPT-type invariant for oriented pseudo links.
result The constructed invariant satisfies a natural pseudo skein relation and admits a state-sum formulation.
An important step for the computation of the HOMFLYPT skein module of the lens spaces via braidsmath.GT
Proves method to compute HOMFLYPT skein module of lens spaces from solid torus.
problem Computing HOMFLYPT skein module of lens spaces from solid torus.
method Solving infinite system of equations involving Lambropoulou invariant and braid band moves.
result Derives HOMFLYPT skein module of lens spaces from solid torus.
Associated with each oriented link is the two variable Homflypt polynomial. The Morton-Franks-Williams (MFW) inequality gives rise to an expression for the Homflypt polynomial with MFW coefficient polynomials. These MFW coefficient polynomials are labelled in a braid-dependent manner and may be zero, but display a numb…
In this paper we study properties of the Markov trace and the specialized trace on the Yokonuma-Hecke algebras, such as behaviour under inversion of a word, connected sums and mirror imaging. We then define invariants for framed, classical and singular links through the trace ${\rm tr}_{d,…
We compare the invariant for classical knots and links defined using the Juyumaya trace on the Yokonuma-Hecke algebras with the HOMFLYPT polynomial. We show that the two invariants, as maps on the set of oriented link types in , do not coincide except in a few trivial cases.
3D topological models link to HOMFLYPT homology via braids.
problem Understanding topological invariants of links and braids.
method 3D topological B-models with Hilbert schemes of points.
result Hilbert space of braids corresponds to HOMFLYPT homology.
Survey of knot invariants in lens spaces.
problem Comparing knot invariants in lens spaces.
method Summarized various invariants and compared their distinguishing power.
result Invariants generalize classical knot polynomials and can distinguish difficult cases.
Given any oriented link diagram, two types of new knot invariants are constructed. They satisfy some generalized skein relations. The coefficients of each invariant is from a commutative ring. Homomorphisms and representations of those rings define new link invariants. For example, the HOMFLYPT polynomial with three va…
Link invariants fail to detect most links with high probability.
problem Detecting specific link types using invariants.
method Mathematical proof and big-data analysis.
result Link invariants have a zero probability of detecting alternating links.
New two-parameter algebra yields stronger link invariants.
problem Developing new link invariants.
method Introducing a two-parameter bt-algebra and its associated invariant.
result The new invariant is more powerful than existing ones.
Polyak showed that any Milnor's -invariant of length 3 can be represented as a combination of Conway polynomials of knots obtained by certain band sum of the link components. On the other hand, Habegger and Lin showed that Milnor invariants are also invariants for string links, called -invariants. We sho…
New knot invariants created using multiple skein relations.
problem Creating more powerful knot invariants.
method Introducing new ways to smooth crossings and using a system of skein equations.
result A simplified version of the modified invariant is easier to compute and generalizes the two-variable Kauffman polynomial.
Researchers compute the skein module of a solid torus using braids.
problem Computing the HOMFLYPT skein module of .
method Braid-theoretic techniques to solve an infinite system of equations.
result The free part of the skein module is generated by the empty link, with all other elements being torsion.
Researchers compute HOMFLYPT skein module of lens spaces using braids.
problem Computing the HOMFLYPT skein module of lens spaces .
method Using braids to solve an infinite system of equations.
result Established the relation between and .
J.-B. Meilhan and the second author showed that any Milnor -invariant of length between 3 and can be represented as a combination of HOMFLYPT polynomial of knots obtained by certain band sum of the link components, if all -invariants of length vanish. They also showed that their formula do…
New algebraic invariants for singular links discovered.
problem Defining and proving invariants for singular links.
method Introduced tied links and tied braid monoid, proved decomposition, re-proved Alexander and Markov theorems.
result Introduced new five-variable polynomial invariants for tied singular links.
The worldsheet skein D-module and basic curves on Lagrangian fillings of the Hopf link conormalmath.SG
Worldsheet skein D-module for Hopf link conormal uniquely determines partition functions.
problem Understanding HOMFLYPT polynomials and their geometric origins.
method Defining worldsheet skein module and D-module, considering skein valued open curve counts.
result Worldsheet skein D-module for Hopf link conormal is generated by three operator polynomials.
New invariant for tied links connects states without resolution dependence.
problem Understanding the Kauffman-like states for tied links.
method Defined Aicardi-Juyumaya states and showed their contribution to the invariant is independent of resolution.
result The double bracket of a tied link diagram can be computed and used to find linked but differently polynomial tied links.
Researchers develop a new basis to simplify solving infinite systems for HOMFLYPT skein module of lens spaces.
problem Computing the HOMFLYPT skein module of lens spaces via braids.
method Using a new basis of the HOMFLYPT skein module of the solid torus, relating infinite systems of equations obtained by performing braid band moves on elements in and via a map .
result Reduced complexity in solving the infinite system of equations for using braids.
New skein invariants defined for links using a novel procedure.
problem Defining new skein invariants for links.
method Applying skein relation to distinct component crossings, then evaluating resulting knots with given invariants.
result Generalizations of known skein invariants for classical links.
This article provides an overview of relative strengths of polynomial invariants of knots and links, such as the Alexander, Jones, Homflypt, Kaufman two-variable polynomial, and Khovanov polynomial.
We find the degree of the Hilbert polynomial for HOMFLYPT homology.
problem Understanding the growth and structure of HOMFLYPT homology.
method Analyzing the degree of the Hilbert polynomial for closed braids.
result The degree of the Hilbert polynomial is for a braid with components.
Introduced coloring-allowed invariants of planar knotoids with the coloring number.
problem Construction of polynomial invariants of knotoids with signs of crossings.
method Defined coloring-allowed invariants of planar knotoids with the coloring number.
result Discussed the 4-phases functions of coloring-allowed invariants.
Aicardi's invariant is extended to colored singular links using graphical calculus.
problem Constructing an invariant for colored classical and singular links.
method State-sum model using graphical calculus for oriented, colored, 4-valent planar graphs.
result Extends to colored singular links, showing it's stronger than HOMFLY-PT polynomial.